Connection to spherical harmonics. Homogeneous spaces.
Risi Kondor Columbia University. Preliminary list of topics:. The use of algebraic methods -- specifically group theory, representation theory, and even some concepts from algebraic geometry -- is an emerging new direction in machine learning. The purpose of this tutorial is to give an entertaining but informative introduction to the background to these developments and sketch some of the many possible applications.
The tutorial is intended to be palatable by a non-specialist audience with no prior background in abstract algebra. CRC Press, IJCAI, In general, the tensor product of irreducible representations is not irreducible; the process of decomposing a tensor product as a direct sum of irreducible representations is known as Clebsch—Gordan theory. Representation theory is notable for the number of branches it has, and the diversity of the approaches to studying representations of groups and algebras.
Although, all the theories have in common the basic concepts discussed already, they differ considerably in detail. The differences are at least 3-fold:. Group representations are a very important tool in the study of finite groups. Over a field of characteristic zero , the representation of a finite group G has a number of convenient properties. First, the representations of G are semisimple completely reducible. This is a consequence of Maschke's theorem , which states that any subrepresentation V of a G -representation W has a G -invariant complement. An irreducible representation of G is completely determined by its character.
Maschke's theorem holds more generally for fields of positive characteristic p , such as the finite fields , as long as the prime p is coprime to the order of G. When p and G have a common factor , there are G -representations that are not semisimple, which are studied in a subbranch called modular representation theory. Hence any G -representation is unitary. Unitary representations are automatically semisimple, since Maschke's result can be proven by taking the orthogonal complement of a subrepresentation.
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When studying representations of groups that are not finite, the unitary representations provide a good generalization of the real and complex representations of a finite group. Results such as Maschke's theorem and the unitary property that rely on averaging can be generalized to more general groups by replacing the average with an integral, provided that a suitable notion of integral can be defined.
This can be done for compact topological groups including compact Lie groups , using Haar measure , and the resulting theory is known as abstract harmonic analysis. Over arbitrary fields, another class of finite groups that have a good representation theory are the finite groups of Lie type. Important examples are linear algebraic groups over finite fields.
The representation theory of linear algebraic groups and Lie groups extends these examples to infinite-dimensional groups, the latter being intimately related to Lie algebra representations. The importance of character theory for finite groups has an analogue in the theory of weights for representations of Lie groups and Lie algebras.
Representations of a finite group G are also linked directly to algebra representations via the group algebra F [ G ], which is a vector space over F with the elements of G as a basis, equipped with the multiplication operation defined by the group operation, linearity, and the requirement that the group operation and scalar multiplication commute. Modular representations of a finite group G are representations over a field whose characteristic is not coprime to G , so that Maschke's theorem no longer holds because G is not invertible in F and so one cannot divide by it. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics , such as algebraic geometry , coding theory , combinatorics and number theory.
A major goal is to describe the " unitary dual ", the space of irreducible unitary representations of G. For non-compact G , the question of which representations are unitary is a subtle one. Although irreducible unitary representations must be "admissible" as Harish-Chandra modules and it is easy to detect which admissible representations have a nondegenerate invariant sesquilinear form , it is hard to determine when this form is positive definite.
An effective description of the unitary dual, even for relatively well-behaved groups such as real reductive Lie groups discussed below , remains an important open problem in representation theory. It has been solved for many particular groups, such as SL 2, R and the Lorentz group. The duality between the circle group S 1 and the integers Z , or more generally, between a torus T n and Z n is well known in analysis as the theory of Fourier series , and the Fourier transform similarly expresses the fact that the space of characters on a real vector space is the dual vector space.
Thus unitary representation theory and harmonic analysis are intimately related, and abstract harmonic analysis exploits this relationship, by developing the analysis of functions on locally compact topological groups and related spaces. A major goal is to provide a general form of the Fourier transform and the Plancherel theorem.
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This is done by constructing a measure on the unitary dual and an isomorphism between the regular representation of G on the space L 2 G of square integrable functions on G and its representation on the space of L 2 functions on the unitary dual. Pontrjagin duality and the Peter—Weyl theorem achieve this for abelian and compact G respectively. Another approach involves considering all unitary representations, not just the irreducible ones. These form a category , and Tannaka—Krein duality provides a way to recover a compact group from its category of unitary representations.
If the group is neither abelian nor compact, no general theory is known with an analogue of the Plancherel theorem or Fourier inversion, although Alexander Grothendieck extended Tannaka—Krein duality to a relationship between linear algebraic groups and tannakian categories.
Harmonic analysis has also been extended from the analysis of functions on a group G to functions on homogeneous spaces for G. The theory is particularly well developed for symmetric spaces and provides a theory of automorphic forms discussed below. A Lie group is a group that is also a smooth manifold. Many classical groups of matrices over the real or complex numbers are Lie groups. The representation theory of Lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply.
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The finite-dimensional representations of G closely correspond to those of K. A general Lie group is a semidirect product of a solvable Lie group and a semisimple Lie group the Levi decomposition. A Lie algebra over a field F is a vector space over F equipped with a skew-symmetric bilinear operation called the Lie bracket , which satisfies the Jacobi identity. Lie algebras arise in particular as tangent spaces to Lie groups at the identity element , leading to their interpretation as "infinitesimal symmetries". Lie algebras, like Lie groups, have a Levi decomposition into semisimple and solvable parts, with the representation theory of solvable Lie algebras being intractable in general.
A representation of a semisimple Lie algebra g is analysed by choosing a Cartan subalgebra , which is essentially a generic maximal subalgebra h of g on which the Lie bracket is zero "abelian". The representation of g can be decomposed into weight spaces that are eigenspaces for the action of h and the infinitesimal analogue of characters. The structure of semisimple Lie algebras then reduces the analysis of representations to easily understood combinatorics of the possible weights that can occur.
There are many classes of infinite-dimensional Lie algebras whose representations have been studied.
Among these, an important class are the Kac—Moody algebras. These algebras form a generalization of finite-dimensional semisimple Lie algebras , and share many of their combinatorial properties. This means that they have a class of representations that can be understood in the same way as representations of semisimple Lie algebras. Affine Lie algebras are a special case of Kac—Moody algebras, which have particular importance in mathematics and theoretical physics , especially conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, Macdonald identities , which is based on the representation theory of affine Kac—Moody algebras.
Lie superalgebras are generalizations of Lie algebras in which the underlying vector space has a Z 2 -grading, and skew-symmetry and Jacobi identity properties of the Lie bracket are modified by signs. Their representation theory is similar to the representation theory of Lie algebras. Linear algebraic groups or more generally, affine group schemes are analogues in algebraic geometry of Lie groups , but over more general fields than just R or C.
In particular, over finite fields, they give rise to finite groups of Lie type. Although linear algebraic groups have a classification that is very similar to that of Lie groups, their representation theory is rather different and much less well understood and requires different techniques, since the Zariski topology is relatively weak, and techniques from analysis are no longer available.
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Invariant theory studies actions on algebraic varieties from the point of view of their effect on functions, which form representations of the group. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant , under the transformations from a given linear group. The modern approach analyses the decomposition of these representations into irreducibles.
Invariant theory of infinite groups is inextricably linked with the development of linear algebra , especially, the theories of quadratic forms and determinants. Another subject with strong mutual influence is projective geometry , where invariant theory can be used to organize the subject, and during the s, new life was breathed into the subject by David Mumford in the form of his geometric invariant theory.
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